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Topic: A good probability puzzle but what is the right wording?
Replies: 10   Last Post: Feb 24, 2013 12:19 PM

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quasi

Posts: 10,259
Registered: 7/15/05
Re: A good probability puzzle but what is the right wording?
Posted: Feb 4, 2013 8:10 AM
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Paul wrote:
>
>The following puzzle is copied and pasted from the internet.
>
> Alice secretly picks two different real numbers by an unknown
> process and puts them in two (abstract) envelopes. Bob
> chooses one of the two envelopes randomly (with a fair coin
> toss), and shows you the number in that envelope. You must
> now guess whether the number in the other, closed envelope
> is larger or smaller than the one you?ve seen. Is there a
> strategy which gives you a better than 50% chance of guessing
> correctly, no matter what procedure Alice used to pick her
> numbers?


Yes.

Let R denote the set of real numbers and let (0,1) denote
the open interval from 0 to 1.

Let f : R -> (0,1) be a strictly decreasing function.

Use the following strategy:

If the initially exposed value is t, "switch" with probability
f(t) and "stay" with probability 1 - f(t).

Suppose Alice chooses the pair x,y with x < y (by whatever
process, it doesn't matter). After Alice choose that pair,
then, by following the strategy I specified above, the
probability of guessing the highest card is exactly

(1/2)*f(x) + (1/2)*(1 - f(y))

which simplifies to

1/2 + f(x) - f(y)

and that exceeds 1/2 since f is strictly decreasing.

Of course, it's not the case that probability of guessing
correctly is more than c for any fixed c > 1/2, but the
problem didn't require that.

quasi



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